Note that, if \(M\)  is finitely generated \(R\) -module, then it follows \(-dim_R(M)\le \operatorname{\textnormal {CI-dim}}_R(M)\le \operatorname{pd}_R(M)\) , and \(\operatorname{\textnormal {CI-dim}}_R(M)<\infty \)  if \(R\)  is a complete intersection ring. Moreover, if \(R\)  is local and \(\operatorname{\textnormal {CI-dim}}_R(M)<\infty \) , then one has by [1]} that
\(-dim_R(M)=\operatorname{\textnormal {CI-dim}}_R(M)=\operatorname{depth}R-\operatorname{depth}_{R}(M).\qquad \mathrm {(\ref {HD}.1)}\) 

